with the initial values s0
= 3, s1 = 0, s2 = 2. This sequence was discussed by
Edouard Lucas in 1878 (American Journal of Mathematics, vol 1, page 230ff),
who noted that if p is a prime then p divides sp. This is an
immediate consequence of Fermat's Little Theorem, and as such is a necessary
but not sufficient condition for primality (see proof). Subsequently (1899) the same
sequence was mentioned by R. Perrin (L'Intermediaire Des Mathematiciens). The
most extensive (published) treatment of this sequence was given in an
excellent paper by Dan Shanks and Bill Adams in 1982 (Mathematics of
Computation, vol 39, n. 159). Shanks and Adams referred to this as Perrin's
sequence.

This sequence has many
interesting properties, making it, in some ways, more remarkable than the
Fibonacci sequence. For example, most people are familiar with the spiral of
equilateral squares whose edge lengths correspond to the Fibonacci numbers,
but less well-known is the spiral of equilateral triangles shown below described
in Spiral Tilings and Integer Sequences.

The edge lengths of successive
triangles in this spiral satisfy the Perrin recurrence sn = sn-2 + sn-3 as well as the recurrence sn
= sn-1 + sn-5, as is apparent from the above
figure. This can be seen as a consequence of the fact that the characteristic
polynomial of Perrin's sequence, x3-
x - 1, is a divisor of the
characteristic polynomial of the 5th order sequence, i.e.,

The real root r of x3- x -
1 has the nice expression as a sequence of nested cube roots:

If we define the angle q in terms of this root r as

then the terms of the Perrin
sequence can be expressed in closed form as

In fact, with the appropriate
provisos, we can replace n with any complex argument z to define Perrin's
function on the complex plane, as shown in the figure below for real
components from -5 to 15 and imaginary components from -5 to +5.

White indicates regions where
both the real and imaginary components are positive, black where both are
negative, and the two shades of gray where one is positive and the other
negative. The zeros of this function are all on the real axis in the left
half-plane, whereas there are two sets of complex conjugate zeros on the
right half-plane (at points where all four shades meet).

Obviously, for any positive
prime p and any integer n we have

so in particular we have

In addition, for any integers
m,n,k we have the relation

Which is really just a special
case of a much more general class of relations satisfied by any linear
recurring sequence. (See the note Identities for
Linear Recurring Sequences.) In this particular case we have relations
like

and so on, as well as

Using these relations for s2n
and s2n+1 gives an efficient means of computing sk via
the usual binary pattern algorithm on the plus and minus sides of the
sequence. This same two-sided approach can be extended to higher order recurrences,
but it quickly becomes more practical to use the more general one-sided
relations (described in the note mentioned above).

Perrin's sequence also has the
interesting property that its terms are cumulative sums of the sequence
itself, i.e., we have s1 = 0 and

The terms of Perrin's sequence
can also be expressed as a function of binomial coefficients, leading to many
interesting results. For example, it's possible to deduce that the summation

As a compositeness test,
Perrin's sequence is much stronger than the typical 2nd order Lucas sequence.
For example, the smallest symmetric pseudoprime relative to the Fibonacci
quadratic x2 – x – 1 is 705, whereas the smallest relative to
Perrin's cubic x3 – x – 1 is

as found by Shanks and Adams
(using an HP-41C calculator). Subsequently Shanks, G. Kurtz, and H. Williams
tabulated all the symmetric pseudoprimes relative to Perrin's sequence less
than 50(10)9. They noted that none of these pseudoprimes had the signature
of a prime p such that Perrin's polynomial is irreducible (mod p). As far as
I know the question of whether such a pseudoprime exists is still open.

Further discussion of
pseudoprimes relative to various polynomials can be found in the notes